Separation of Variables Across ODEs, Heat PDEs, and Eigenfunction Methods

All flashcards

How can a first-order ODE be recognized as separable?

Its right-hand side factors as $g(x)h(y)$, so the variables can be separated before integration.

How does the substitution $z=y^{(n-1)}$ help with some higher-order ODEs?

For $y^{(n)}=f(y^{(n-1)})g(x)$, it gives the separable first-order equation $z'=f(z)g(x)$.

What role do boundary conditions play in separated heat-equation solutions?

They select the permissible spatial eigenvalues and eigenfunctions.

What ODE describes each mode of a forced one-dimensional heat equation?

For mode $n$, it is $u_n'+\alpha\frac{n^2\pi^2}{L^2}u_n=h_n(t)$.

What is a Kronecker sum of two matrices?

For matrices $A$ and $B$, it is $A\otimes I+I\otimes B$, with identity matrices of appropriate sizes.

Quiz questions

1. For an ODE of the form $y^{(n)}=f(y^{(n-1)})g(x)$, what equation results from setting $z=y^{(n-1)}$?Answer: $z'=f(z)g(x)$, a first-order separable equation.
2. For the heat equation $u_t=\alpha u_{xx}$ on $0\le x\le L$ with $u(0,t)=u(L,t)=0$, which eigenvalues allow nontrivial spatial modes?Answer: $\lambda_n=\frac{n^2\pi^2}{L^2}$ for $n=1,2,3,\ldots$
3. For the sine-series expansion of initial data $f(x)$ on $[0,L]$, what is the coefficient of $\sin(n\pi x/L)$?Answer: $D_n=\frac{2}{L}\int_0^L f(x)\sin\left(\frac{n\pi x}{L}\right)\,dx$
4. For a forced heat equation with zero endpoint conditions, which equation governs mode $n$?Answer: $u_n'(t)+\alpha\frac{n^2\pi^2}{L^2}u_n(t)=h_n(t)$
5. How is a two-dimensional discrete Laplacian assembled from one-dimensional operators $D_{xx}$ and $D_{yy}$?Answer: $D_{xx}\otimes I+I\otimes D_{yy}$